Burning the Cosmic Commons:
Evolutionary Strategies for Interstellar Colonization
Robin Hanson∗
Department of Economics
George Mason University†
July 1, 1998

Abstract
Attempts to model interstellar colonization may seem hopelessly compromised by uncertainties regarding the technologies and preferences of advanced civilizations. If light speed limits
travel speeds, however, then a selection eﬀect may eventually determine frontier behavior. Making weak assumptions about colonization technology, we use this selection eﬀect to predictcolonists’ behavior, including which oases they colonize, how long they stay there, how many
seeds they then launch, how fast and far those seeds ﬂy, and how behavior changes with increasing congestion. This colonization model explains several astrophysical puzzles, predicting loneoases like ours, amid large quiet regions with vast unused resources.

∗
For their comments, I thank Curt Adams, William Bainbridge, Forrest Bishop, Nick Bostrom, Russell Brand,
Alexander Chislenko, Wei Dai, Carl Feymnan, Hal Finney, Sandy Hodges, Al Globus, Geoﬀrey Landis, Eugene
Leitl, Tim May, Peter McCluskey, Ralph Merkle, Perry Metzger, William Newman, James Nolan, Anders Sandberg,
Richard Schroeppel, Damien Sullivan, Nick Szabo, Frank Tipler, Ed Turner, Dan Werthimer, and participants of
the University of West Virginia economics department seminars. This paper was inspired by a comment by Carl
Feynman that a selection eﬀect might determine asymptotic frontier colonization behavior.
†rhanson@gmu.edu http://hanson.gmu.edu 704-993-2326 FAX: 704-993-2323 MSN 1D3, Carow Hall, Fairfax VA
22030

1

1

Economics and the Search for Extraterrestrial Intelligence

For many decades, the search for extraterrestrial intelligence (ETI) has proceeded using advanced
astronomical techniques (Cosmovici, Bowyer, & Werthimer, 1997). In contrast, the social theory
which directs such empirical attentions, suggesting where such ETI may be distributed and thekinds of observable activities it may be engaged in, has been relatively primitive.
Most research in this area has been done by astronomers, who have given careful theoreticalattention to the distribution of planets like ours near stars like ours. Sophisticated biological theory
has also been applied to the question of where life such as we see on Earth could arise, travel, andsurvive (Lawler, 1998). Some attention has even been given to how often simple life may give birth
to a technologically advanced civilization like humanity’s.
Regarding the possible activities of advanced civilizations, sophisticated engineering has been
applied to place bounds on the feasibility and costs of interstellar travel (Crawford, 1995) and onlarge visible engineering projects such as intercepting most of the light of a star, or even disassembling planets and stars (Criswell, 1985). Sophisticated communication theory and astrophysics
has also been applied to suggest low cost methods for an ETI to communicate across astronomical
distances.
However, when it comes to considering which physically possible activities advanced civilizations
might choose to engage in, the search for ETI has been mostly informed by astronomer’s intuitive
theories of the nature of societies. (There are a few notable exceptions (Finney, 1985; Wachter,
1985; Bainbridge, 1984).)
For example, a striking empirical puzzle regarding ETIs is the lack of alien visitors here on
Earth (Brin, 1983), even though ETI does not seem terribly hard to evolve, the universe is very
big, and ETI civilizations could have been expanding for billions of years. Attempts at explanations
include our star system being part of an isolated zoo (Ball, 1973), or being within an empty borderzone between civilizations (Turner, 1985).
Also, while simple population diﬀusion processes have been frequently used to model the expansion of ETI civilizations (Jones, 1976, 1995; Newman & Sagan, 1981; Bainbridge, 1984; Fogg, 1987),
crucial features of such models are typically based on simple intuitive arguments. For example,
some feel it is the nature of life to ﬁll all available niches, while others see advanced civilizations as
having overcome base expansionist tendencies.
Those who lean toward niche-ﬁlling tend to assume fast cheap interstellar travel, high per-capita
wealth, and population growth and emigration rates no less than that of recent human history(Jones, 1995; Barrow & Tipler, 1986). This implies fast expansion rates for ETI civilizations, andhence suggests that the lack of alien visitors here now implies that ETI is very rare.

2

Others, including the most prominent astronomers who discuss ETI, tend to see interstellar
travel as slow and expensive, per-capita wealth as limited (Drake & Sobel, 1992), and recent humanpopulation trends as historical abberations (Newman & Sagan, 1981). Such assumptions can imply
slow expansion rates, allowing our galaxy to be full of ETIs while Earth is untouched. This can
support optimism in the search for ETI. For example, Harvard astronomer Michael Papagiannis
argues that
The limits of growth in a ﬁnite system, which will be imposed on all stellar civilizations
by the colossal distances that separate the stars, will aﬀect the natural selection ofthese civilizations. Those that manage to overcome their innate tendencies toward
continuous material growth and replace them with non-material goals will be the only
ones to survive this crisis. As a result the entire Galaxy in a cosmically short period
will become populated by stable, highly ethical and spiritual civilizations. (Papagiannis,
1984)
Astronomers Carl Sagan and William Newman are even willing to assume zero-population
growth, which implies a very slow rate of expansion (Newman & Sagan, 1981). They reason
Those civilizations devoted to territoriality and aggression and violent settlement ofdisputes do not long survive after the development of apocalyptic weapons. Civilizations
that do not self-destruct are pre-adapted to live with other groups in mutual respect.
This adaptation must apply not only to the average state or individual, but, with very
high precision, to every state and every individual within the civilization ... the result
is that the only societies long-lived enough to perform signiﬁcant colonization of the
Galaxy are precisely those least likely to engage in aggressive galactic imperialism.
(Sagan & Newman, 1983)
Since the search for ETI seems to be an extreme version of the proverbial search for a needle ina haystack, theoretical expectations are important in guiding empirical searches for ETI. And since
the largest theoretical uncertainties seem to regard the social aspects of ETI, it seems desirable
to replace astronomer’s intuitive social theories with more sophisticated social theory regarding
ETI. In particular, formal economic theory may be both general enough to hope to apply to theextreme circumstance of ETI societies, and formal enough to allow a clearer identiﬁcation of whichassumptions lead to which conclusions.
Economic theory has been applied to the consideration of human institutions and policy regarding space for a while now (Banks, Ledyard, & Porter, 1989; Butler & Doherty, 1991); perhaps
it is time to consider a wider scope for economics in space studies. An economic analysis of interstellar colonization might not only inform the search for ETI, it may also inform us about the3possible futures for our distant descendants, and may serve as a colorful base for generalizations toanalogous spatial colonization processes. Such analogous processes include colonization by humans
in the recent (Ely, 1918; Luthy, 1961) or remote past, by other biological organisms both large
and small, and by agents colonizing more abstract spaces, such as companies colonizing spaces ofpossible concepts and designs.

2

Overview of Results

This paper oﬀers an economic style analysis of the consequences of a wave of colonization having
expanded from some source for long enough, and in a decentralized enough manor, that a selectioneﬀect determines frontier behavior. That is, we assume suﬃcient diversity or mutability of colonist
values, suﬃcient autonomy for colonies to express those preferences in diﬀering colonization strategies, and suﬃcient spacing between the origins of colonization waves to allow the leading edge ofcolonization to come to be dominated by a selection eﬀect: leading edge colonists must travel at
the maximum sustainable average speed.
While these are not the only reasonable assumptions one could make, they are among thosecommonly considered, and this paper describes some relatively general implications of these assumptions. In contrast to the population-biology based models previously used to discuss ETI
expansion, which implicitly make speciﬁc behavioral assumptions such as speciﬁc crude correlations between emigration actions, local population levels, and the motion of a colonization wave
(Murray, 1993; Newman, 1980, 1983), we here explicitly consider strategic actors who may choose
from a wide range of behaviors.
Speciﬁcally, we consider a wave of colonization, wherein seeds land at oases, grow colonies there,
and then send out more seeds, each of which travels much far than the oasis spacing. We assume
that property rights in virgin oases are not enforced, so oases go to whoever arrives ﬁrst. (Property
rights seem especially hard to enforce regarding leading-edge seeds traveling at a substantial fractionof the speed of light.) We also assume that this process has persisted long enough that frontier
behavior is dominated by a selection eﬀect: there are no other behaviors which would allow a
population of seeds to sustain or grow their number while pulling ahead of the overall colonization
wave. (Selection eﬀects have been observed in simulations of interstellar colonization across evenrelatively small scales (Bainbridge, 1984), and have been considered as explanations for preferences
and behaviors in a wide variety of economic contexts (Hansson & Stuart, 1990; Blume & Easley,1992).)In this analysis, we allow the behavior of a seed or colony to be contingent on its locationin space-time. As random factors make colonists fall back or jump ahead relative to the overall
4

colonization wave, colonists adjust their behavior in light of their preferences regarding creating
descendants at various points in space-time, and their expectations about the equilibrium behaviorof other colonies. Colonists are free to choose the types of oases they land at, how long they staythere, the speed, hardness, and perception abilities of seeds, and how far those seeds ﬂy before
they attempt to colonize a new oasis. Assuming all colonies are risk-neutral and have the same
preferences and technology, we derive constraints on such equilibrium preferences and strategies.
Since eventually most of the volume of the colonized universe should be far behind the colonization frontier, we are particularly interested in the preferences and behavior of colonists who havefallen far behind the frontier, perhaps left behind at an oasis after its main seed exodus.
We consider ﬁrst the behavior at the very leading edge of colonization, then consider behaviors
elsewhere in the colonization process. We then consider behaviors in a traveling wave of colonization,
where far from the colonization origin the process has settled down to a stable one-dimensional
pattern moving through space. Finally, using particular functional forms, we obtain closed form
expressions for a smooth approximation, and we discuss how these results generalize when there
are a diversity of oasis types. We get the following results.
At the leading edge, great growth at oases is matched by great death between oases; on average
only one of all the seeds launched from an oasis survives to colonize a new oasis. Also, a great
premium is placed on seed hardness; a trillion plain seeds are worth as much as a million seeds
twice as penetrating, which are worth as much as a thousand seeds which penetrate twice as muchagain.
Everywhere in the colonization wave, succeeding generations stay longer at oases before sendingout seeds, and place a smaller premium on being closer to the leading edge. A large premium iseverywhere placed on seed hardness, as the ideal elasticity between seed cost and seed hardness is
at least the total seed mortality factor.
As congestion increases, making it diﬃcult for seeds to ﬁnd unoccupied oases, seeds attempt
landings at more and more oasis types. Even with diverse oasis types, however, all seeds launchedfrom a given space-time region have the same speed and hardness, and target the same destination
space-time region. Comparing nearby oases, there is some cutoﬀ growth rate such that oases withthat cutoﬀ rate launch seeds, and colonies which are growing faster wait longer. If seed launching
reduces the local growth rate, colonists left behind after a seed exodus will want to leave as soon
as possible.
In a traveling wave of colonization, individual seeds always ﬂy faster than the speed of thecolonization wave, and colonists have inﬁnite patience regarding investments whose returns keep
up with that traveling wave. Succeeding generations of colonies fall further behind the leading edge,
increase in number, and per colony deliver more seeds to each succeeding space-time region. With5

ﬁnite oases and a minimum mortality rate for seeds in ﬂight, there is an upper limit to the fractionof oases which get colonized, and a lower limit to the oasis growth rate colonists will tolerate. Whenseed hardness is ﬁxed, each generation sends out more seeds at a slower speed, which have a lower
mortality rate at the end of their ﬂight.
If the preferences of the very ﬁrst colonists were the preferences that would eventually be selected
for, or if early colonists directly valued having their descendants selected for, this sort of analysis
might inform the strategies of the earliest colonists, perhaps including when our descendants should
leave our solar system. We ﬁnd two cases where we can show that seeds should not be launched until
after local growth rates slow down: anywhere in a traveling wave, and at any generalized leadingedge when seeds suﬀer a constant mortality between oases. This suggests that selected-preferencecolonists will not leave our solar system until after economic growth rates reverse their historical
trend so far toward increasing growth rates.
Regarding the possibility that an ETI colonization wave has already passed this way, we cancompare this theoretical analysis with astrophysical observations now considered relevant to thesearch for ETI. This includes the “Great Silence” (Brin, 1983); no alien ETI appear on Earthnow, no strong radio signals are being steadily sent here from nearby stars, and astrophysicists
have had a great and increasing success at explaining astrophysical phenomena in terms of simple
physical processes, without reference to ETI societies. For example, we can see that any civilizations
surrounding nearby stars dump little nuclear waste into their stars (Whitmire & Wright, 1980),
don’t use powerful radars (Tarter, 1997), don’t use certain forms of starship drives (Drake & Sobel,
1992), and intercept less than one percent of their star’s light (Jugaku & Nishimura, 1997).
What does the model in this paper suggest a long-colonized region should look like? Surprisingly,
this model allows for lone oases like ours amid large regions apparently devoid of advanced activity,and containing large amounts of unused resources. Very few colonists should remain far behindthe frontier, and any stragglers should be in a big hurry to catch up with the leading edge, rather
than to communicate with us, or to take on slow resource extraction projects which might result
in large visible structures.
On the other hand, the model also suggests that we should a wall of colonization at some,perhaps large, distance from here. And we should observe an cutoﬀ in the distribution of growth
rates that observed resources can support, with an absence of resources supporting rapid growththat simple physical processes led one to expect to be more common. We may also observe garbage
artifacts where the missing oases once were, and the remains and exhaust of seeds. Finally, themodel oﬀers no obvious reason for such colonists to take any trouble to hide their activities fromus or anyone else. More work seems required to see if this model is consist with our astrophysicalobservations.
63

Assumptions and Notation

Think of space as mostly nearly-empty space with a few exceptional “oases” such as stars. The
resources available to a colony at an oasis depend on how long that colony has been growing
there. Assuming there is eﬀectively only one interesting resource dimension, and let the increasing
function R̃(s) describe the total resources available to a colony which as been growing at an oasis
for a duration s. (Section 15 summarizes all our notation.)
Note that this form R̃(s) implicitly assumes that oases are identical across space and time, andthat technology is unchanging. We thus neglect the diversity of star types, and the overall evolutionof the universe. Oasis diversity will be examined in section 12. Unchanging technology seems aplausible assumption in the limit when physical laws become well-known and a very wide space oftechnological designs becomes well explored.
Resources are required to launch a new seed with speed v and hardness h. Assume that all seeds
launched from a given oasis have the same speed v and hardness h. (This assumption will follow
from an assumption of risk-neutrality made in section 5. Let F (v, h) be the ﬁxed resource cost tobuild a seed launching facility, and let C(v, h) ≥ 1 be the constant marginal cost to launch one more
seed. If all seeds launched from an oasis are launched at the same time, then N = (R̃(s) − F )/C
is the number of seeds launched per oasis. (Assumptions which imply identical seed launch dates
are given in section 11.)
Seeds traveling between oases may fail to reach their destination due to internal failures, ordue to collisions with dust grains or cosmic rays. Let mortality A(x, v, h) ≥ 1 be the number ofseeds with speed v and hardness h which would have to be launched to expect one to survive on
average after traveling a distance x. (Note that this form assumes a uniform unchanging inter-oasis
medium, neglecting for example the clumping of dust around galaxies.) An noteworthy referencecase is constant mortality, where A is exponential in x, so there is a constant probability per unit
distance of a still-surviving seed being destroyed.
Let r, t denote a space-time event at a radius r and time t from the center and beginning ofa spherically-symmetric colonization process. For x the minimum distance seeds travel between
oases, and n the number density of oases, if x  n−1/3 then we can ignore the detailed locations ofoases and just consider P (r, t), the fraction of oases near r that are occupied at time t. The rate
at which oases are becoming occupied is Pt (r, t) ≥ 0.
Assume that at some point a seed must commit to a choice of an oasis it will attempt tocolonize, and that if this attempt fails, the seed is too weak to make a second attempt at anotheroasis. Assume further that seeds have imperfect information about which oases are currently
occupied, and about the existence and intentions of other seeds in ﬂight. Since physical limits on

7

travel speeds make it hard to see how one could enforce property rights in virgin oases, we assume
that the ﬁrst seed to land at an oasis gets it, and we neglect any variations in costs to defend oases
against new seeds.
Given these assumptions we can focus on the local probability of a successful landing
Q(r, t; v, h) = q(P (r, t), Pt (r, t); v, h),where q(P, Pt ; v, h) gives the probability of landing at an unoccupied oasis when a fraction P ofoases are occupied, when occupancy is increasing at a rate Pt , and when the seed has speed v andhardness h. A speciﬁc binary-signal model for q(P, Pt ; v, h) will be described in section 8.
Assume that, aside from the problems of in-ﬂight mortality and already occupied oases, seeds
can reliably detect and land at suitable oases. Then since it is always possible to randomly choose
target oases, we must have Q ≥ 1 − P . We also assume that P = 1 implies Q = 0, since one cannot
colonize if all oases are taken. We will also typically want to assume increasing congestion, with
Qt ≥ 0 and Qr ≤ 0.
The distance between generations of colonies is x, the distance a seed travels, and the time
between generations is
T = s + δ(v, h) + x/v,
where s is the time between oasis colonization and seed launch, v is seed velocity, and δ(v, h) is thetime delay required for a seed to stop at a new oasis.
Given that one seed successfully colonizes an oasis at r, t, the expected number of seeds which
can be delivered to the event r + x, t + T can be written as generational growthG(x, T ; v, h) =R̃(T − δ(v, h) − x/v) − F (v, h)N=.
A
C(v, h) A(x, v, h)

The expected number of successfully colonized oases at r + x, t + T , however, isG(x, T ; v, h) Q(r + x, t + T ; v, h).
To simplify our model, we will from here on neglect the dependence of delay, ﬁxed cost, andoasis detection on v and h. That is, we assume δv = δh = Fv = Fh = qv = qh = 0. Thus we candeﬁne an increasing net resource function R(T − x/v) = R̃(T − δ − x/v) − F . We will also assumeA(x, v, h) = A(b(v, h)x), for A an increasing function, so that the seed mortality function retains
its shape as seed velocity and hardness varies. Thus we write8

G(x, T ; v, h) =R(T − x/v)
C(v, h) A(b(v, h)x)(1)

and we subsume δ in s, writing s = T − x/v.
For any function f (z), let us write lnf (z) = ln(f (z)). Then lnG is locally convex in x, T if andonly if lnR is concave and lnA is convex (i.e., lnA ≥ 0 ≥ lnR ), so that the in-ﬂight mortality rate
lnA increases with distance and the oasis growth rate lnR decreases with time. Note that evenwith a concave total resource growth function lnR̃, the delay δ and ﬁxed cost F terms can induce
non-concave regions in the net resource function lnR(s) = lnR̃(x − δ) − F , where lnR > 0. We
will typically assume, however, that lnR is concave at the actual chosen s.

4

At The Leading Edge

At the very leading edge of colonization, P (r, t) approaches zero, and so Q(r, t) must approachone. Assume for the moment that it were regularly possible for the descendants of a seed to growin number while remaining along some candidate “leading edge”, so that for r, t and r + x, t + T
both “at” the leading edge, G(x, T ; v, h) > 1. Then it would be possible for seeds at this leadingedge to grow a little slower while pulling ahead of this leading edge, via G(x, T − ; v, h) > 1or G(x + , T ; v, h) > 1. If any “leading edge” colonists took this strategy and did not die out
completely, then eventually their descendants should become numerous enough that P (r, t) would
not approach zero at the candidate “leading edge”. And so it would not actually be the leadingedge.
If a leading edge is to be stable against this sort of selection eﬀect, it must not be possible togrow there regularly. That is, the strategies x, T, v, h at a stable leading edge must satisfyxsuch that G(x, T ; v, h) ≥ γ,
x,T,v,h T
max(2)for γ = 1. Call the value of this maximum speed w, and call the arguments which produce thismaximum x̃, T̃ , ṽ, h̃. (Naturally, s̃ = T̃ − x̃/ṽ also.)
If the constraint G ≥ 1 binds, we have G = 1 or A = N . This is our ﬁrst noteworthy result.
Result 1 At a stable leading edge, on average only one of all the seeds launched from an oasis
survives to colonize a new oasis.
For any γ we can write the FOC (ﬁrst order condition) for equation 2 with respect to x, T as
w=

lnGT
x̃
=−
lnGx
T̃
9

(3)

and with respect to v, h as 0 = Gv = Gh .
Plugging in our expression for G (equation 1), the FOC with respect to x, T becomesb lnA
s̃
=
,
x̃
lnR

(4)

and the SOC (second order condition) becomes
lnA
lnR
≤
.
lnR2
lnA2
Note that this SOC implies that for constant mortality, where lnA = 0, we must have lnR < 0.
Result 2 With constant mortality, leading edge colonists don’t launch seeds until after local net
growth rates start to slow down.
To the extent that the earliest colonization eﬀorts, near r = t = 0, can be thought of as roughly
following leading edge strategies, satisfying equation 2 for some value of γ, under conditions of nearconstant seed mortality, then colonists should not leave the colonization origin (such as our solar
system) in earnest until growth rates begin to slow down there.
Constant mortality is also a useful reference case for considering seed hardness h. With constant√mortality, a doubling of seed hardness via b → b/2 makes A → A, for which one is willing to pay√
up to C → C A when attempting to min C A. That is, for A = N one is willing to raise seed costs√
so much that N → N , holding and R constant.
Result 3 With constant mortality at the leading edge, a trillion plain seeds are worth a million
seeds twice as penetrating, which are worth a thousand seeds which penetrate twice as much again.
Note that the same value ratios apply to any innovation which everywhere halves local mortality√
rates d lnA/dx, since that also makes total A → A.

5

Behind The Leading Edge

The ﬁnite oasis spacing, together with small variations in seed launch times and speeds, should
induce noise regarding the seed-landing events r + x, t + T of the children of a seed which landed at
x, T . The accumulation of such noise will eventually cause some leading edge seeds to fall behindthe leading edge of colonization, moving into space-time regions with substantial population P > 0
and congestion Q < 1. Since eventually most the volume of the colonized universe is in regions
occupied by fallen-behind colonists, we are particularly interested in the strategies and preferencesof such colonists.
10To ﬁnd such strategies and preferences, we will assume that the induced value U (r, t) to colonistsof having a seed successfully land at an oasis at event r, t is given by a strategy equation
U (r, t) = max G(x, T ; v, h) Q(r + x, t + T ) U (r + x, t + T ),x,T,v,h(5)

the expected number of child seeds which successfully land at unoccupied oases near the new event,times the value of having seeds at that new event.
Note that this form assumes risk-neutral colonists and neglects to explicitly consider noise
regarding the landing event r + x, t + T . The reasonableness of these approximations are discussed
in sections 9 and 10. Note also that risk-neutrality by itself implies that it is always optimal for all
the seeds launched from a given oasis to have the same strategy x, T, v, h.
Let x(r, t), T (r, t), v(r, t), h(r, t) be the arguments which maximize the strategy equation (5), andfor any function f (r, t) of space-time, let us write f¯(r, t) = f (r + x(r, t), t + T (r, t)), the function fapplied to the succeeding generation after r, t. The strategy equation (5) then implies
U (r, t) = G(x, T ; v, h)Q̄(r, t)Ū (r, t).

(6)

The local growth rate of the colonist population is given by a ﬂow equation
rP̄t (r, t) = Pt (r, t) G(x, T ; v, h) Q̄(r, t)
r+x

−1
2 


det ∂(r + x, t + T )  ,∂(r, t)(7)

where the last two terms correct for the divergence across space-time of colonist strategies. Notethat the form of this equation implicitly assumes a uniform distribution of oases.
The strategy and ﬂow equations (5 and 7) are the primary equations of our model, being
suﬃcient for example to numerically compute a colonization equilibrium. It turns out, however,that almost all our results will come from considering the strategy equation alone.
The FOC of the strategy equation (5) with respect to x, T , together with expressions forlnUr , lnUt , imply

lnUr = − lnGx = − lnḠx + lnQ̄r ,(8)lnUt = − lnGT = − lnḠT + lnQ̄t .

(9)Since GT ≥ 0 ≥ Gx , colonists discount returns that clearly fall behind the colonization wave, withUr ≥ 0 ≥ Ut . If we also assume increasing congestion, Qr ≥ 0, Qt ≤ 0, then − lnGx ≥ − lnḠx ≥ 0
and lnGT ≥ lnḠT ≥ 0, so that lnUr , and − lnUt are positive and decreasing with successive11

generations.
Result 4 The premium placed on being closer to the leading edge decreases with succeeding generations.
Substituting for G via equation 1, we ﬁnd lnGT = lnR , so equation 9 and lnR ≤ 0 implys̄ ≥ s.
Result 5 Succeeding generations stay longer at oases before sending out seeds.
The FOC for our strategy equation (5) with respect to v, h are 0 = Gv = Gh . The maximization
with respect to hardness h is equivalent to minh C A(bx). The FOC for this can be written in termsof the optimal elasticity between seed cost and seed hardness as−

lnCh= bx lnA (bx).
lnbh

(10)

The right-hand side of equation 10 is the number of probes that would have to be launched
for just one to survive had the ending mortality rate applied over the whole distance. For an
increasing mortality rate, lnA > 0, this is greater than the total mortality factor A. Thus the
optimal elasticity between seed cost and hardness can be very large.
Result 6 The ideal elasticity between seed cost and seed hardness is at least the total seed mortality
factor.
The FOC for speed v is
lnCv /x = lnR /v 2 − bv lnA .

6

(11)

A Traveling Wave

The diﬀerential equations which are typically used to describe spatial growth and combustion for
physical and biological systems in uniform media typically have traveling wave solutions (Murray,1993), and typically a wide range of initial conditions asymptotically approach such traveling wave
solutions. In a traveling wave in one spatial dimension, the wave retains the same shape as it moves
along the medium. Since the above model may asymptotically approach such a traveling wave, it
is of interest to look for such a solution.
That is, let us focus attention far away from the colonization origin, where on the scales ofinterest planes of similar behavior should make the process essentially one-dimensional. And let us
12consider the possibility that for r  x the main functions f (including P, Q, x, T, v, h) can be writtenf (r, t) = f (r − ω(t − t0)) = f (e), where ω is the velocity of the traveling wave, e(r, t) = r − ω(t − t0 )is the extremity of a space-time event relative to the traveling wave center. Thus, with only a slightabuse of notation, we can write fr = f  and ft = −ωf  (where f  = f  (e)).
For a traveling wave, the ﬂow equation (7) becomes(1 + ∆e )P̄  (e) = P  (e) G(x(e), T (e); v(e), h(e)) Q̄(e),(12)

where for any f we write ∆f = f¯ − f , the diﬀerence between succeeding generations. The motionof succeeding generations relative to a traveling wave is ∆e = x − ωT , so ∆e > 0 for a lineage that
advances relative to the wave, and ∆e < 0 for a lineage that falls behind the wave.
Assuming leading-edge colonists just keep up with the leading edge, then 0 = ∆e = x̃ − ω T̃
there, so we must have ω = w, with the wave speed being equal to the fastest sustainable leadingedge speed from equation 2. If there were another place e behind the leading edge where ∆e (e) = 0
and Q(e) < 1, then by equation 12 we would have G(e) > 1, and so w would not be the fastest
sustainable leading edge speed at Q = 1. Thus there is no other place where ∆e = 0, and sobehind the leading edge we either have ∆e < 0 everywhere, or ∆e > 0 everywhere.
Since in a traveling wave lnQ̄t = −w lnQ̄r , the strategy FOC with respect to x, T (equations 8, 9)
imply


∆

lnGx
lnG̃x





=∆

lnGT
lnG̃T



,

(13)

where lnG̃x , lnG̃T are the leading edge values. Since the ratios whose change equation 13 considers
are equal at the leading edge, and since equation 13 says these ratios change by the same amount,these ratios must remain equal for any colony which has some ancestor or descendant at the leadingedge. Thus assuming that all colonies have such a leading edge ancestor or descendant, they must
all satisfy the same equation 3 which the leading edge must satisfy, and so
−w =

lnUt
lnGT
=
.
lnGx
lnUr

(14)

Equation 14 implies U (r, t) = U (e), so there is no discounting along the traveling wave.
Result 7 Colonists have inﬁnite patience regarding investments whose returns keep up with a traveling wave.
Though our model has assumed ﬁxed technology, this result may be relevant to future models
which consider colonists attitudes toward investing in research.
13

Given U (r, t) = U (e), the strategy equation (5) for a traveling wave implies
U (e) = G(x(e), T (e); v(e), h(e)) Q̄(e)Ū (e),and equation 8 implies U  > 0. Thus ∆e ≥ 0 implies Ū ≥ U , and hence GQ ≤ 1, with adeclining number of descendants everywhere. Since this is not consistent with any sensible boundary
condition on the ﬂow of colonists at r = 0, we must have instead ∆e ≤ 0 everywhere, and henceŪ ≤ U and GQ ≥ 1.
Result 8 In a traveling wave, succeeding generations of colonies on average fall further behind theleading edge and increase in number.
In a traveling wave, the SOC for the strategy equation (5) with respect to x, T can be written∀α,0≥∂ 2 lnG(1 − αw)2∂ 2 lnU=+lnU  ,∂(x + αT )2∂(x + αT )2
1 + ∆e(15)

where we expect 1 + ∆e to remain positive to avoid a singularity in the ﬂow equation (12), andwherelnU  =lnGT T T 2 − lnGxx x2b2 lnA + lnR (T 2 − x2 /v 2 )=.
x + wT x + wT 

Since increasing congestion and equation 8 imply lnŪ  ≤ lnU  , we havelnU  > 0(16)

on average over each range [e−∆e(e), e]. If lnU  > 0 holds at each point, and if the SOC equation 15
is satisﬁed, we must have lnG strictly convex, which implies lnA ≥ 0 ≥ lnR .
Result 9 In a traveling wave, seeds are only launched when oasis growth rates are slowing down,
and seeds only land when mortality rates are rising.
Thus to the extent that the earliest colonization eﬀorts, near r = t = 0, can be thought of as
roughly following traveling wave strategies, colonists should not leave the colonization origin (suchas our solar system) in earnest until growth rates begin to slow down here.
If we substitute our expression for G (equation 1) into equation 14, we ﬁnd
1
lnA
1
= +b
,
w
v
lnR

14

(17)

Assuming positive growth and mortality, lnR , lnA > 0, equation 17 immediately implies thatv > w.
Result 10 In a traveling wave, seeds always ﬂy faster than the wave speed.
Let us deﬁne best growth G̃(x, T ) = maxv,h G(x, T ; v, h) and consider the trajectory of succeeding generations in x, T space. Starting with some generation at the leading edge, and so at
the point x̃, T̃ on the T = wx line, succeeding generations move away from this point while thegradient of lnG̃ remains orthogonal to this line (by equation 14) and the magnitude of this gradientdecreases with succeeding generations (by equations 8 and 9).
If we assume lnG is strictly locally convex in x, T , then so is lnG̃, which implies that along the
curve of ﬁxed gradient direction, the gradient must be monotonic in lnG̃. Thus growth G increases
with succeeding generations.
Result 11 In a traveling wave, each colony expects to delivers more seeds to its succeeding spacetime region than its parent expected to deliver.
Equation 17 implies that if there is a lower bound to mortality b lnA and an upper bound to
seed speed v̂ (such as the speed of light), then there is also a lower bound to oasis growth rates
lnR ≥ lnR = b lnAv̂w.
v̂ − w

Since GQ > 1 and G = R/CA, we have Q > CA/R, so using our assumptions C ≥ 1 and A ≥ 1,
an upper bound on oases resources R̄ implies a limit on congestion,
Q ≥ Q = 1/R̄.
And since P = 1 implies Q = 0, there must also be an upper limit P ≤ P̄ < 1 to the fraction ofoases which are colonized.
Result 12 With ﬁnite oases and a minimum mortality in a traveling wave, there is an upper limit
to the fraction of oases which get colonized, and a lower limit to the oasis growth rate colonists will
tolerate.
If one ﬁxes seed speed and hardness v, h, then equation 17, s̄ ≥ s, and locally convex lnG implyx̄ ≤ x, which implies that if lnGv = 0 initially, then for children lnḠv ≤ 0. Thus if we ﬁx only
seed hardness h, allowing speed to vary, we must have decreasing speed v̄ ≤ v, which given Cv ≥ 0
implies C̄ ≤ C, so that number of seeds N = R/C must increase. Equation 17 further implies that
end-of-ﬂight mortality b lnA = (lnA(bx))x must decrease.
15

Result 13 In a traveling wave with ﬁxed seed hardness, each generation sends out more seeds ata slower speed, which have a lower mortality rate at the end of their ﬂight.

7

Smooth Traveling Wave In a Linear ModelIf colonist strategies x, T, v, h do not change much from generation to generation, one might hope
to approximate this model by a local diﬀerential equation.
For a traveling wave, equation 8 implies
w lnQ̄ = lnR − lnR̄ = −the approximation being accurate when lnR s̄slnR (s)ds ≈ − lnR ∆s,

2 lnR /∆s. Similarly, when lnA

2 lnA /b∆x,

and we ﬁx seed speed and hardness v, h, then we have (1 − w/v) lnQ̄ ≈ −b lnA ∆x. These
approximations are exactly correct for linear mortality & growth, where lnR = lnA = 0.
These approximations describe expressions for ∆s, ∆x in terms of lnQ̄ and lnR , lnA , which
given ∆e = (1 − w/v)x − ws imply
w2(1 − w/v)2−∆(∆e)=
K≡+.
− lnR
b2 lnAlnQ̄
We can further approximate −K lnQ̄ = ∆(∆e) ≈ ∆e ∆e = 12 ((∆e)2 ) when∆e2∆e /∆e.

(18)

If we assume linear mortality & growth, then K is constant. Using the leading edge boundary
condition that Q̄ = 1 at ∆e = 0, i.e., that colonists at the leading edge tend to stay there, we thengetQ̄ = exp(−

(∆e)2
),
2K

(19)

an explicit Guassian functional relation between Q̄ and ∆e. And equation 14 can be rewritten[− lnR (1111− )]s + [b2 lnA ]x = [lnR (0)( − ) − b lnA (0)],
w v
w v

(20)

where all bracketed quantities are constants. Equations 19 and 20, together with∆e = (1 − w/v)x − ws

16

(21)

and the boundary values x̃, s̃, now completely describe the relationship between all parametersQ̄, ∆e, x, s (assuming ﬁxed v, h) in a smooth traveling wave for linear mortality & growth.
The only thing we haven’t described is how any one of these parameters varies explicitly withposition e, where a colony sits relative to the leading edge. That requires making use of the
population ﬂow equation 12.
Result 14 For ﬁxed seed hardness and speed, and linear mortality & growth in a slowly-varying(equation 18) traveling wave, equations 19, 20, 21 give explicit relationships between how longcolonists wait at oases s, how far they ﬂy x, how much each generation falls behind ∆e, and thechance seeds land at an open oasis Q̄.

8Modeling Congestion

We have deﬁned congestion Q in terms of q(p, pt ), which gives the probability of landing at anunoccupied oasis when a fraction p of oases are occupied, when occupancy is increasing at a rate
pt . We now consider an explicit binary-signal model for q.
One expects colonies and seeds to study possible target oases, and perhaps communicate withother nearby seeds in ﬂight regarding landing intentions. Assume that at the time they must
commit to a choice of target, seeds consider a binary signal regarding whether an oasis is open (◦)
or closed (•), i.e., available to colonize or will have been taken already.
Let q◦ be the conditional probability of getting an open signal given an open oasis, and let q•
be the conditional probability of a closed signal given a closed oasis. A seed which then randomly
chooses among open-signal oases then has a chance Q to land at an open oasis, given by
P
1−Q
=
,
Q
1−P
where error  = (1 − q• )/q◦ ≤ 1. This satisﬁes Q ≥ 1 − P and P = 1 implies Q = 0, as we haveassumed.
While our model has so far not allowed for a choice of error , if such a choice were allowed, an
optimal choice would


min C

P
1+
1−P



,

which for C < 0 implies that C increases with P .
Result 15 The fewer open oases remain, the more seeds spend to discern which oases remain open.
17

If the diﬃculty of detecting colonized oases depends on how long those oases have been colonized,then the expression for  is modiﬁed. Assume that a recently colonized oasis, roughly less thanτ old, has diﬀerent signal correlations q̂◦ , q̂• . Then with a fraction D ≈ τ pt of recently-colonizedoases, we have=

(1 − q• )(1 − D) + D(1 − q̂• ).
q◦ (1 − D) + Dq̂◦Note that with this form, if Pt is not monotonic in P , then Q need not be monotonic in P , and soincreasing population Pt ≥ 0 need not imply increasing congestion Qt ≤ 0.

9

Risk-Aversion

The form of the strategy equation (5) assumes risk-neutral colonists. It seems reasonable to assume
that the risks of seed mortality are largely independent across widely separated seeds, and that
there are large numbers of widely separated seeds with copies of any one “gene” coding for colonist
behavior. Given these assumptions, any plausible level of risk aversion for a gene should translate tonear risk-neutrality regarding the risks to any one seed (Blume & Easley, 1992). Thus risk-neutrality
should be a reasonable approximation, at least regarding large numbers of widely separated seeds.
Risk neutrality is a more suspect approximation regarding risks which are strongly correlated
over large regions, such as the risk of a gamma ray burst, or that this colonization wave will hit
an opposing wave coming from another source. These possibilities have not been modeled in thispaper.
Note also that risk-neutrality seems a bad approximation when mortality is exactly constant.
To see this, consider any strategy, starting at r, t, of waiting s1 and then going x1 , followed by
waiting s2 and then going x2 , in both cases using the same v, h. For risk-neutrality and constantmortality, this sequence is dominated by the strategy of waiting s1 , going x1 + x2 , waiting s2 andthen going x = 0, at least if by waiting one encounters less congestion, as inQ(r + x1 , t + s1 + x1 /v) < Q(r + x1 + x2 , t + s1 + (x1 + x2 )/v).
The waiting strategy has exactly the same total growth at oases, R1 and R2 , and given constantmorality has the same total mortality between oases A1 A2 , but has better congestion Q. That is, aseed would always prefer to ﬂy farther to land rather than to suﬀer more congestion from landing
earlier. But if all seeds do this, none would ever land, a nonsensical result.
Result 16 Risk-neutrality is a bad approximation when mortality is exactly constant.
18

10

Noisy Landings

Note also that the form of equation 5 neglects to explicitly include noise regarding landing events.
A form which included noise would look something like


max
x,T

G(. . .) Q(. . .) U (r + x + x , t + T + T )dF (x , T ; x, T ).

The form we have used is, however, a reasonable approximation to this form when noise is small.
To see this, consider maximizing some V (z) when z has noise distributed by F (), and assuming0=dF () and 0 = V  = V  . . .. Then the error from ignoring noise is∆z = argmaxz= S(1 −≈
where σ 2 =

σ
S



 

σ
2

V (z + )dF () − argmaxz V (z)1 − (σ/S)2 ) for σ ≤ S
for σ 2S2,

 2
 dF () is the standard deviation of noise, and S = −V  /V  evaluated at argmaxz V (z)is the distance from this point where the SOC, V  ≤ 0, no longer holds. Thus for σ∆z

S, we haveσ, a small correction indeed.

Result 17 Our analysis seems robust to substantial noise in where and when seeds actually land.

11

Seed Diversity

We have so far assumed that all seeds launched from an oasis are launched at the same time. Let
us now consider allowing more diversity in seek launching dates.
Think of an oasis as made of a variety of resources, some of which permit faster growth thanothers. Some asteroids, for example, may combine good element compositions with good access tostarlight, while others may have combine poor starlight access with deﬁcits of certain key elements.
We expect colonies to begin growing in the best resources, and to switch to less desirable resources
as the best resources become depleted.
Assume that a colony can at any time be described by it’s working capital K(s) and the resources
R(s) that it has so far consumed, and that the colony begins with K(0) = R(0) = 1. The colony
can consume resources at a rate R = Kg(R), where g(R) is the capital growth rate possible after
R resources have been consumed. The colony output R can be devoted to either further capital

19

growth K  or to sending out seeds. Previously accumulated capital K can also be devoted tosending seeds.
If the value of a launched seed declines with the launch date according to V (s), the colony
should thus want to choose K(s) to maximize
 ∞
0





K(s)g(R(s)) − K  (s) V (s)ds(22)

subject to R(0) = K(0) = 1, K ≥ 0, and K  (s) ≤ K(s)g(R(s)) = R (s). Before considering thisproblem directly, however, let us consider the related problem where growth rates are ﬁxed withtime: maximize
 ∞
0





K(s)g(s) − K  (s) V (s)ds(23)

subject to K(0) = 1, K ≥ 0, and K  (s) ≤ K(s)g(s).
The Euler-Lagrange ﬁrst order condition for problem 23 isg(s) = γ(s),where we have deﬁned γ(s) = − lnV  (s), a local discount rate. This equation says that for interior
optima, the local growth rate must equal the local discount rate. Thus if g(0) > −γ(0), we must
initially have a boundary solution where K  (s) = K(s)g(s), with all resources devoted to capital
accumulation. If feasible growth rates decline relative to discount rates, until a time t∗ when suchrates become equal, with g(s∗ ) = γ(s∗ ), then this all-capital boundary solution will continue untils∗ , at which time capital can be spent on sending out seeds. If g (s) < γ  (s) after s∗ as well, thenthere must be a boundary solution after s∗ , with K = 0. Thus all capital is spent on seeds at s∗ .
If we now consider g(R(s)) instead of g(s), we can ask whether this solution to problem 23 also
describes a solution to problem 22. That is, let us deﬁne R̄ as the maximal capital accumulation
growth path which solves R̄ (s) = R̄(s)g(R̄(s)) and R̄(0) = 1, let us analogously assume thatg(1) > γ(0), that g (R̄(s))R̄ (s) ≤ 0 ≤ γ  (s), and that for some s∗ we have g(R̄(s∗ )) = γ(s∗ ). Now
consider whether it is optimal to accumulate capital along K(s) = R̄(s), and then spend it all
sending seeds at s∗ .
There are two possible deviations to consider: sending seeds out before s∗ or afterward. If some
capital is spent sending seeds early, at a time s1 < s∗ , this deviation is a net win only if the added
gain from getting an earlier seed outweighs the loss from delaying when the other seeds can belaunched. That is, if

20V (s1 ) − V (s∗ ) > V  (s∗ )R̄(s∗ ) R̄(s∗ )R̄(s1 )−dR,
g(R)R2where the integral describes how the ideal seed sending time s∗ varies with earlier capital K(s1 ).
This condition is impossible to meet given our assumption that growth rates are faster than discountrates over this period. An intuition for this is aided by noting that with constant growth anddiscount rates, i.e., g(R) = g and γ(s) = γ, the deviation win condition becomes(eγ∆s − 1)/γ > (eg∆s − 1)/gfor ∆s = s∗ − s1 , which contradicts our assumption that γ ≤ g in this region.
The other deviation to consider is to save and not spend some capital at s∗ , growing it instead
to send a seed at some later date. Our assumption of increasing discount rates, γ  > 0, ensuresthat this capital will grow at a rate g(R(s)) ≤ g(R̄(s∗ )) ≤ γ(s). Hence the growth rate would beless than the local discount rate, giving a net loss for this deviation.
Our traveling wave analysis, however, suggested decreasing, not increasing, discount rates (seeequation 16). And in this situation it is optimal to leave some capital behind, and to continue tosave enough capital to maintain the relation g(R(s)) = γ(s). The optimal capital fraction, however,isγ  (s)/γ
d lnγ
K
=
=

R
R g (R)
ds

dg
,
d lnR

which is very small when discount rates vary slowly with time and growth rates vary substantiallywith resources R. This suggests that almost all seeds are sent in the main wave at s∗ , making it
reasonable approximation to assume in our main analysis that all seeds are sent at the same time.
Result 18 When local growth rates decline as more resources are consumed, and when local discount rates vary slowly, almost all seeds are launched at the same time.
Even assuming that the small-fraction of “left-behind” colonists has a negligible eﬀect on therest of our analysis, however, such colonists are of particular interest in their own right. After all,most of the volume of the colonized universe should eventually be far behind the leading edge, andthis volume may be dominated by such left-behind colonists. Besides strategic reasons, colonists
may be left behind due to random factors, or due to being embodied in ﬁxed sunk capital such as
seed launchers.

21

12

Oasis Diversity

We have so far required all oases to have the same resource function R(s), but most of our results
generalize to the case where resources Rβ (s), and hence growth Gβ depend on the oasis type β.
Assume that seeds have no trouble distinguishing between oases of diﬀerent types, though they
may still have trouble determining if an oasis is colonized. With oasis diversity, leading edgecolonists will then seek out the most favorable oasis type, and so the leading edge maximum speed
equation (2) will also maximize over β, choosing some best value β̃. Our previous results about theleading edge apply regarding this type of oasis β̃.
If seed costs do not depend on the type of oasis they are to land at, then if landings are being
attempted at more than one type of oasis in any space-time region, colonists must be indiﬀerent
between landing at these oasis types. Assume that landings are always being attempted at the best
type β̃ if they are being attempted at any type, and for any f let the unsubscripted version be forthat best type, as in f = fβ̃ . ThenQβ (r, t) Uβ (r, t) = Q(r, t) U (r, t).
must be satisﬁed for all β where Pt (β) > 0. An oasis type for which seed landings are locally being
attempted for the ﬁrst time must have Qβ = 1, and so all else equal, the lower Q = Qβ is, theworse a Uβ /U ratio can be tolerated for a type β.
Result 19 All else equal, greater congestion induces seeds to attempt landings at more oasis types.
An analogue to the strategy equation (5) must apply for each colonized oasis type,Uβ (r, t) = max Gβ (x, T ; v, h) Q(r + x, t + T ) U (r + x, t + T ).
x,T,v,hand so the same form of FOC, equations 8 and 9, must be satisﬁed for all such oasis types. Thus
all oases who try to land seeds at the same space-time region must have the same values of lnGx
and lnGT , hence the same value of lnRβ , and hence exactly the same equations determining x, v, h.
Result 20 Even with diverse oasis types, all seeds launched from a given space-time region havethe same speed and hardness, and target the same destination space-time region.
Consider the situation where in some small region some oases are launching seeds now whileother oases are still waiting to launch, and still others have already launched. All the launchingoases have the same cutoﬀ growth rate lnR̂ . When local growth rates slow with time, lnRβ < 0,22

then if another nearby colony is waiting longer, it must be because it is growing faster than this
cutoﬀ rate, lnRβ > lnR̂ , while a colony that has already launched must have expected to have alower-than-cutoﬀ growth rate had it not launched.
Thinking in terms of multiple oasis types can help clarify intuitions regarding the behavior ofleft-behind colonists. A post-exodus oasis can just be thought of as an already-colonized oasis ofa new type β. If the exodus process decreases local growth rates, then left-behind colonists would
want to launch more seeds as soon as possible, rather than staying to grow further. (In the modelin the previous section (11), exodus left growth rates unchanged.)
Result 21 All local oases with some cutoﬀ growth rate launch seeds, while faster growing oases
wait longer. If seed launching decreases local growth rates, left-behinds want to leave A.S.A.P.

13

Observational Implications

We have tried in this paper to model colonists who descend from leading edge colonists, and who
have retained their ancestor’s preferences, all far away from the origin of an expanding colonization
wave. One possible application of this model is to understand our descendants’ future, assumingthat they eventually engage in interstellar colonization, that light speed remains a fundamental
limit to travel speeds, and that alien ETI are suﬃciently distant that there is time for selectioneﬀects to determine frontier behavior.
Another possible application of this model is to try to reconcile the “Great Silence” we seein astrophysics with a hypothesized previous wave of ETI colonization that might have long ago
passed this way. How consistent are the implications of this model with astrophysical observation?

13.1

Puzzling Observations

No known astrophysical phenomena seem puzzling if one assumes that any one star system is
extremely unlikely to give birth to an expanding wave of interstellar colonization. If simple life
is extremely unlikely to develop in any one system, or if simple life were extremely unlikely togenerate a civilization powerful enough to begin interstellar colonization, then one can believe thatover the ten billion plus years our universe has existed, none of the over 1020 star systems we can
now see were the origin of a colonization wave that passed this way. In this section, however, we
focus on reconciling our astrophysical observations with more optimistic estimates of the chances
for originating powerful civilizations.
The lack of alien visitors here on Earth now is striking (Brin, 1983), but upon reﬂection is not
terribly surprising. Even if the vast majority of star systems in our galaxy have been colonized23

by ETI, there are many reasons to expect a few exceptions. Our star system might be part of an
isolated zoo (Ball, 1973), for example, or in an empty border zone (Turner, 1985).
Our failure to so far detect strong radio communications between ETIs may also not be terribly
surprising. After all, we have sampled only a very small fraction of the space of not-obviously
unreasonable communication system parameters. Our failure to detect communications directed at
us may perhaps be more surprising, if we assume Earth is of interest to ETIs. For example, when
we looked at 209 nearby stars like our sun, we failed to hear anything like the equivalent of Earth’s
currently most powerful transmitter (Arecibo Radar) talking to us. At the ten nearest such stars,
we failed to hear any transmitter 500 times weaker (Dreher & Cullers, 1997).
On reﬂection, perhaps the most puzzling astrophysical observation is the vast quantities ofapparently unused resources out there. If one thought most stars were colonized, and if one expectedmature stellar civilizations to make use of a substantial fraction of the resources near their star,then one might be surprised to learn that a survey of 230 observed nearby stars like our sun foundthat less than one percent of the starlight of each is being intercepted by local matter (Jugaku &Nishimura, 1997) (at least matter that reradiates near 300K). This seems a striking contrast to
some predictions, such as by Barrow and Tipler (Barrow & Tipler, 1986), of a complete exhaustionof material resources in the wake of a colonization wave.
The great success of stellar physics also indicates that very few stars have been much modiﬁed
by local civilizations, such as to extract mass. And we observe vast quantities of apparently unusedmaterial in molecular and dust clouds. Such apparently virgin mass may seem puzzling given thaneven now we can sketch out engineering projects which might make use of such resources (Criswell,1985). We can also see that civilizations surrounding nearby stars dump very little nuclear waste
into their stars (Whitmire & Wright, 1980), and we can place limits on the use of powerful radars(Tarter, 1997) and certain forms of starship drives (Drake & Sobel, 1992).

13.2

Reconciling the Puzzles

The model in this paper suggests a way to reconcile the hypothesis of a colonization wave having
once passed this way with observations of apparently low activity levels and resource utilization
nearby. The model implies low local activity levels, as there should be few colonists left far behindthe colonization frontier. The model also implies that any such colonists would be in a rush totry to catch up with the frontier, which does not oﬀer any obvious motivation to broadcast signals
at Earth. And the model accounts predicts a small residue of untouched oases, such as our solar
system seems to be.
The model might also explain observations of unused starlight, stellar mass, and other mass24sources, if these resources cannot support rapid growth. We mainly need to assume that large
engineering projects, such as required to collecting most of the light of a star or to extract much ofits mass, just take too long to produce an acceptable rate of return. This might explain both theunused resources and the absence of large visible engineering projects.
While the model can explain these previously puzzling observations, it also predicts a numberof new phenomena that we have not yet observed, nor failed to observe. Future observations whichlook more carefully for such phenomena will thus oﬀer opportunities to test this model.
First, the model predicts that a “wall” of fresh colonization frontier activity should be observable
at some distance, though that distance could be very far if the frontier passed this way a long timeago. Given a speciﬁcation of oasis growth and in-ﬂight mortality functions, the model even predicts
how thick this wall should be, and how fast it would be moving. How visible the wall should be,
however, depends on how much radiation is produced by fresh colonization activities.
Second, the model predicts an absence of local resources which can support rapid growth. If
for any given resource we can estimate the most rapid possible growth rate possible using the best
technological designs, then we should see a cutoﬀ in the distribution of such growth rates, acrossthe types of resources observed. This value of this cutoﬀ should be consistent with a fuller modelof the colonization process, including a wave speed and rates of in-ﬂight mortality. And modelsof the physical processes which produce the distribution of resources available should predict thegeneration of fast-growth resources which we fail to observe nearby.
Third, we should ﬁnd garbage and remains where there were once resources supporting rapid
growth. Up close we might even ﬁnd the remains of any ﬁxed sunk costs such as mines and seedlaunchers, as colonists in this model have no particular reason to want to cover their tracks. The
better colonists were at recycling capital, however, the less such garbage should remain.
Fourth, we may ﬁnd the remains of seeds which suﬀered from in-ﬂight mortality, as well as any
exhaust from in-ﬂight propulsion and navigation. Of course it is possible that most propulsion was
produced in the launching system, and that most of the remains of failed seeds continued on until
colliding with its intended destination system. But some residues may remain to be detected.

13.3

A Further Complication

A further complication is that this model would seem to allow for several colonization waves, at
slower and slower speeds w. After the ﬁrst colonization wave has passed through a region, anotherslower wave might pass through, using oases and their remainders that grow too slowly to interest
the ﬁrst wave. Colonists in slower waves would have substantially diﬀerent preferences and behavior
from ﬁrst-wave colonists. They might originate from the same colonization origin as the ﬁrst wave,25

or from large preference mutations in colonists during ﬁrst wave.
If the ﬁrst wave did not collide with waves from other sources, if the second wave came fromthe same origin as the ﬁrst, and if it had a speed substantially less than 79% of the speed of theﬁrst wave, then the second wave would have reached substantially less than half of the volume of
the ﬁrst wave, and predictions which ignored later waves might remain reasonable. In other cases,
a more complex analysis seems required.
In summary, this model allows our solar system to have been passed by a colonization wave,
and yet be an untouched oasis in a very large region containing much unused resources, and little
advanced activity or large artifacts. But the model also predicts a colonization wave observable atsome distance, an absence of physically-expected fast-growth resources nearby, possibly replaced
by a few garbage artifacts, and perhaps some remains from in-ﬂight seed mortality and propulsion.
More careful astrophysical analysis seems required to more directly confront these predictions withobservations. And a more careful analysis of multiple colonizations waves may be in order.

14

Conclusion

It seems desirable to try to replace astronomer’s intuitive social theories with more formal economic
theory in directing the empirical search for ETI. In that spirit, this paper takes a few commonly
considered assumptions, and given those assumptions considers the interstellar colonization behavior of sophisticated actors with many behavioral choices. The hope is that this theoretical analysis
may be both general enough to hope to apply to the extreme circumstance of ETI societies, andformal enough to allow a clearer identiﬁcation of which assumptions lead to which conclusions.
Far enough away from the origin of an expanding wave of interstellar colonization, and in theabsence of property rights in virgin oases, a selection eﬀect should make leading edge colonists
primarily value whatever it takes to stay at the leading edge. Metaphorically, a ”wildﬁre” would
burn through the ”cosmic commons,” turning most easy kindling into a residue of charcoal andash, but leaving slow-burning trunks mostly untouched. After the ﬁre has past, the rare untouchedsapling would ﬁnd itself in a large quiet region with few competitors.
That is, to the extent that descendants of such colonists who fall behind the leading edge retain
their ancestor’s preferences, such descendants should focus on pursuing the colonization wave as
quickly as possible, rather than staying behind to build stable long-lasting civilizations. While suchcolonists would adjust their strategies to deal with increased congestion, there are limits to have
far their strategies vary from leading-edge strategies. Thus our observations of apparently being alone oasis among a large region almost devoid of large scale advanced activity may be consistent
with a colonization wave having once passed this way.
26

This analysis has neglected to consider many important issues, which will hopefully be addressedby future research. Perhaps most seriously, this analysis has not much considered the possibly ofslower waves of colonists whose preferences diﬀer substantially from those selected-for at the leadingedge of the ﬁrst wave. And we have not very carefully considered non-uniformities in the inter-oasis
medium.
Regarding the behavior of colonists left far behind the leading edge, it seems important to explicitly model attitudes toward risk, the possibility of property rights and other forms of coordination
among colonies, and possible selection-induced drift in colonist preferences.
The other end of the colonization process also seems worthy of more attention. Assuming
one is at the beginning of a colonization process, and that one wants one’s descendants to staycompetitive with the colonization wave, what strategies would that imply? When should theyleave, for example? At this end of the process, it seems especially important to consider risk
aversion and to allow for technological change.

References
Bainbridge, W. S. (1984). Computer Simulation of Cultural Drift: Limitations on Interstellar
Colonization. Journal of the British Interplanetary Society, 37, 420–429.
Ball, J. A. (1973). The Zoo Hypothesis. Icarus, 19, 347.
Banks, J. S., Ledyard, J. O., & Porter, D. P. (1989). Allocating Uncertain and Unresponsive
Resources: An Experimental Approach. RAND Journal of Economics, 20, 1–25.
Barrow, J. D., & Tipler, F. J. (1986). The Anthropic Cosmological Principle. Oxford University
Press, New York.
Blume, L., & Easley, D. (1992). Evolution and Market Behavior. Journal of Economic Theory, 58,
9–40.
Brin, G. D. (1983). The ‘Great Silence’: The Controversy Concerning Extraterrestrial Intelligent
Life. Quarterly Journal of the Royal Astronomical Society, 24, 283–309.
Butler, A. M., & Doherty, N. A. (1991). Torts and Orbits: The Allocation of the Costs of Accidents
Involving Spacecraft. The American Economic Review, 81 (2), 46–49.
Cosmovici, C. B., Bowyer, S., & Werthimer, D. (Eds.). (1997). Astronomical and Biochemical
Origins and the Search for Life in the Universe, No. 161. Editrice Compositori.

27

Crawford, I. A. (1995). Interstellar Travel: A Review. In Extraterrestrials, Where are They?
(Second edition)., pp. 50–69. Cambridge University Press, Cambridge.
Criswell, D. R. (1985). Solar System Industrialization: Implications for Interstellar Migrations. In
Finney, B. R., & Jones, E. M. (Eds.), Interstellar Migration and the Human Experience, pp.
50–87. University of California Press, Berkeley.
Drake, F., & Sobel, D. (1992). Is Anyone Out There? The Scientiﬁc Search for Extraterrestrial
Intelligence. Delacorte Press, New York.
Dreher, J. W., & Cullers, D. K. (1997). SETI Figure Of Merit.. In Cosmovici et al. (1997), pp.
711–717.
Dyson, F. J. (1966). The Search for Extraterrestrial Technology. In Marshak, R. (Ed.), Perspectives
in Modern Physics, pp. 641–655. Wiley, New York.
Ely, R. T. (1918). Private Colonization of the Land. The American Economic Review, 8 (3),522–548.
Finney, B. R. (1985). Voyagers into Ocean Space. In Finney, B. R., & Jones, E. M. (Eds.),
Interstellar Migration and the Human Experience, pp. 164–179. University of California Press,
Berkeley.
Finney, B. R., & Jones, E. M. (1985). Interstellar Migration and the Human Experience. Universityof California Press.
Fogg, M. J. (1987). Temporal Aspects of the Interaction among the First Galactic Civilizations:
The “Interdict Hypothesis”. Icarus, 69, 370–384.
Hansson, I., & Stuart, C. (1990). Malthusian Selection of Preferences. American Economic Review,
80 (3), 529–544.
Jones, E. M. (1976). Colonization of the Galaxy. Icarus, 28, 421–422.
Jones, E. M. (1995). Estimates of Expansion Timescales. In Extraterrestrials, Where are They?
(Second edition)., pp. 92–102. Cambridge University Press, Cambridge.
Jugaku, J., & Nishimura, S. (1997). A search for Dyson spheres around late-type stars in the solar
neighborhood II.. In Cosmovici et al. (1997), pp. 707–710.
Kuiper, T., & Brin, G. D. (1989). Extraterrestrial Civilization. American Association of Physics
Teachers, College Park, MD.
28

Lawler, A. (1998). Ames Tackles the Riddle of Life. Science, 1840–1841.
Luthy, H. (1961). Colonization and the Making of Mankind. Journal of Economic History, 21 (4),483–495.
Murray, J. D. (1993). Mathematical Biology (Second edition). Springer, Berlin.
Newman, W. I. (1980). Some Exact Solutions to a Non-Linear Diﬀusion Problem in Population
Genetics and Combustion. Journal of Theoretical Biology, 85, 325–334.
Newman, W. I. (1983). The Long-time Behavior of the Solution to a Non-linear Diﬀusion Problem
in Population Genetics and Combustion. Journal of Theoretical Biology, 104, 473–484.
Newman, W. I., & Sagan, C. (1981). Galactic Civilizations: Population Dynamics and Interstellar
Diﬀusion. Icarus, 46 (3), 293–327.
Papagiannis, M. D. (1984). Natural Selection of Stellar Civilizations by the Limits of Growth.
Quarterly Journal of the Royal Astronomical Society, 25, 309–318.
Regis, E. (1985). Extraterrestrials, Science and alien intelligence. Cambridge University Press,
New York.
Sagan, C., & Newman, W. I. (1983). The Solipsist Approach to Extraterrestrial Intelligence.
Quarteryly Journal of the Royal Astronomical Society, 24, 113–121.
Tarter, J. (1997). Results from Project Phoenix: Looking Up From Down Under.. In Cosmovici
et al. (1997), pp. 633–643.
Turner, E. L. (1985). Galactic Colonization and Competition in a Young Galactic Disk. In Papagiannis, M. D. (Ed.), The Search for Extraterrestrial Life: Recent Developments, pp. 477–482.
International Astronomical Union.
Wachter, K. (1985). Voyagers into Ocean Space. In Finney, B. R., & Jones, E. M. (Eds.), Interstellar
Migration and the Human Experience, pp. 120–133. University of California Press, Berkeley.
Whitmire, D., & Wright, D. (1980). Nuclear Waste Spectrum as Evidence of Technological Extraterrestrial Civilizations. Icarus, 42, 149–156.

29

15

Glossary

A Number of seeds which must be launched to expect one to survive the ﬂight.
b Parameter controlling the rate at which seeds are destroyed in ﬂight.
β Type of oasis.
C Cost to launch one more seed, assuming the ﬁxed cost has been paid.
δ Time delay required for a seed to stop and begin to grow at an oasis.
D The fraction of oases which have been colonized ”recently”.
e Position of a space-time event relative to a traveling wave.
 The error rate in signals regarding whether oases are still available.
F Fixed cost required to support launching of seeds.
G Expected number of seeds a single grown seed delivers to the next space-time region.
h Hardness of seed.
n Number density of oases in local region of space.
N Number of seeds launched from an oasis.
P Fraction of local oasis that have already been colonized.
q◦ (q• ) The conditional probability of seeing a truthful signal indicating an open (closed) oasis.
Q Probability that a seed will land at an already-colonized oasis.
r Distance from origin of colonization process.
R Resources available to colony at an oasis, net of ﬁxed launching cost.
R̃ Total resources available to colony at an oasis.
s Time duration a colony has been at an oasis, including delay for seed to stop at oasis.
t Time since the start of colonization process
T Time duration between seed generations, from seed landing to seed landing.

30

U Utility of having a seed land at an unoccupied oasis in some space-time region.
v Velocity of seed in ﬂight.
w Average maximum sustainable speed of colonization
ω Average speed of a traveling wave of colonization.
x Distance a seed travels from where it was launched to where it lands.
—————————–
f Some arbitrary function of space and time.
lnf A new function of space time equal in value to the natural logarithm of the old one.
fr The partial derivative of f with respect to location r.
ft The partial derivative of f with respect to time r.
f  The partial derivative of f relative to the wave position along a traveling wave.
∆f The diﬀerence between f for seeds landing here now, and f for the next generation.

31